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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 215B QFT Winter 2025}\\
{\Large\bf Assignment 8 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Tuesday, March 4, 2025}




\bigskip
\hrule
\begin{enumerate}

\item {\bf Bubble-chain approximation for bound states.}


In discussing the form of the spectral density for an operator that creates a massive particle,
I mentioned that in addition to the single-particle delta function at $s=m^2$, and the continuum above $s=(2m)^2$, 
there could be delta functions associated with bound states at $m^2 < s < (2m)^2$.  Here we'll get an idea how we might discover such a thing theoretically.

For this problem, we're going to work in $D=2+1$ dimensions, so that 
we can avoid the problem of UV divergences.  Consider the theory of a single real scalar with action
$$ S[\phi] = \int d^3 x \( \half \partial_\mu \phi \partial^\mu \phi - \half m^2 \phi^2 - { g \over 4! } \phi^4 \) $$
where $m, g$ are real.  In this problem we will consider both signs of $g$, without worrying about questions of the stability of the vacuum (maybe there is a small $\phi^6$ term that saves the day but can be ignored here).


\answer{I got this problem from Lawrence Hall.}


\begin{enumerate}
\item Consider the amplitude $\CM(s)$ for elastic scattering $\phi \phi \to \phi \phi $, with $s = E_T^2$,
the square of the total center of mass energy.
Compute $\CM(s)$ in the bubble-chain approximation, defined as the following infinite sum of Feynman diagrams:
$$ \parfig{.6}{fig-11-large-N-bubble-sum.png} + \cdots $$
Do not worry about justifying the validity of the approximation (it is not justified in this theory, though it is in a large-$n$ version of the theory), and do not worry about convergence of the series.
You can leave your answer as a Feynman parameter integral.


\item Show, by explicit calculation, that the bubble chain approximation to the scattering amplitude obeys the optical theorem.  
[In elastic scattering in the center of mass frame in 3d, the element of solid angle $d\Omega$ is just an element of ordinary angle $d \theta$, and $ d\sigma/d\theta = {|\CM|^2 \over 32 \pi p E_T^2 }$ where $p$ is the magnitude of the spatial momentum of either particle.]




\item The interaction between the $\phi$ quanta could result in two of them forming a bound state of mass $M_B$.  A signal of such a bound state is the appearance of a pole in $\CM(s)$ at $s= M_B^2$ on the real axis, but below threshold ($0<M_B^2 < 4m^2$).  Find the values of $g$ for which the bubble-chain approximation predicts bound states.  
[You are not asked to give an analytic expression for $M_B$.]


\end{enumerate}





\item {\bf Another consequence of unitarity of the $S$ matrix.}

\begin{enumerate}
\item Show that unitarity of $S$, $S^\dagger S = \Ione = S S^\dagger$,
implies that the transition matrix is {\it normal}: 
\be\label{eq:detailed-balance} \CT \CT^\dagger  = \CT^\dagger \CT~.\ee

\item What does this mean for the amplitudes $ \CM_{\alpha\beta}$
(defined as usual by $ \CT_{\alpha\beta} = \bdelta(p_\alpha - p_\beta) \CM_{\alpha\beta}$)?



\item The probability of a transition from $\alpha$ to $\beta$ is 
$$ P_{\alpha \to \beta} = |S_{\beta\alpha}|^2 = VT \bdelta(p_\alpha-p_\beta) | \CM_{\alpha \beta}|^2$$
which is IR divergent.  More useful 
is the transition rate per unit time per unit volume:
$$ \Gamma_{\alpha \to \beta} \equiv {P_{\alpha\to\beta}\over VT}.$$
Show that the 
the total decay rate of the state $\alpha$ is 
$$ \Gamma_\alpha \equiv \int d\beta  \Gamma_{\alpha \to \beta} 
%\buildrel{\text{optical thm}}\over{=} 
= 2 \Im \CM_{\alpha\alpha}.$$




\item Consider an ensemble of states $p_\alpha$ evolving according 
to the evolution rule
\be\label{eq:master} \partial_t p_\alpha = - p_\alpha \Gamma_\alpha + \int d\beta p_\beta \Gamma_{\beta \to \alpha} .\ee
$ S[p] \equiv - \int d\alpha p_\alpha \ln p_\alpha $ 
is the Shannon entropy of the distribution.
Show that $$ {dS \over dt } \geq 0 $$
as a consequence of \eqref{eq:detailed-balance}.
This is a version of the Boltzmann $H$-theorem.





\item{} [Bonus]
Notice that we are doing something weird in the previous part by using classical probabilities.  
This is a special case;
more generally, we should describe such an ensemble by a density matrix $ \rho_{\alpha\beta}$.
Generalize the result of the previous part appropriately.






\end{enumerate}

\item {\bf Abrikosov-Nielsen-Oleson vortex string.}


 Consider the Abelian Higgs model in $D=3+1$:
$$ \CL_h \equiv - {1\over 4 } F_{\mu\nu} F^{\mu\nu} 
+ \half |D_\mu\phi|^2 - V(|\phi|) $$
where $\phi$ is a (complex) scalar field of charge $q$ whose covariant derivative is 
$ D_\mu \phi = \( \partial_\mu - \ii q e A_\mu \) \phi $,
and let's take 
$$V(|\phi|) = { {\kappa\over 2 }}  (  |\phi|^2 - v^2)^2 $$
for some couplings $\kappa, v$.  
Here we are going to do some interesting classical field theory.
Set $ q=1$ for a bit.

\begin{enumerate}

\item Consider a configuration that is independent of $x^3$, one of the spatial coordinates, 
and static (independent of time).
Show that its energy density (energy per unit length in $x^3$) is
$$ U = \int d^2 x ~ \( \half F_{12}^2  + { \half} |D_i \phi|^2 + V(|\phi|) \) .$$



\item{} [optional, but used crucially below] Consider the special case where $\kappa = { \kappa_0  =  \( { e q \over 2  } \)^2 }$.  
Assuming that the integrand falls off sufficiently quickly at large $ x^{1,2}$, show that
$$ U_{\kappa=1} = \int d^2 x ~ \( \half  \( F_{12} +  { \sqrt{\kappa}} \( |\phi|^2 - v^2 \)  \)^2 
+ { {1\over 4 }} | D_i \phi + \ii \eps_{ij} D_j \phi|^2 
+  { \sqrt{\kappa}} v^2 F_{12} - { \half} \ii \eps_{k\ell} \partial_k \( \phi^\star D_\ell\phi\)  \) . $$

%$\varsigma$


\item The first two terms in the energy density of the previous part
are squares and hence manifestly positive, so setting to zero 
the things being squared will minimize the energy density.
Show that the resulting first-order equations (they are called BPS equations
after people with those initials, Bogolmonyi, Prasad, Sommerfeld)\footnote{Let's set $\kappa=1$ for this discussion; it does not affect the qualitative conclusions.} 
$$ 0 =\(  D_i + \ii \eps_{ij} D_j \) \phi, ~~~ F_{12} = - |\phi|^2 + v^2 $$
are solved by ($ x^1 + \ii x^2 \equiv r e^{ \ii \varphi} $)
$$ \phi = e^{ \ii n \varphi} f(r) , ~~A_1 + \ii A_2 = - \ii e^{ \ii \varphi } { a(r) - n \over r } $$
if
$$ f' = { a \over r } f, ~ a' = { r} (f^2 - v^2 ) $$
with boundary conditions 
\be\label{eq:BCatinfty} a \to 0, f \to v + \CO\(e^{ - m r} \) , ~~\text{at } r \to \infty \ee
$$ a \to n + \CO(r^2), f \to r^n (1 + \CO(r^2 ) ), ~~~ \text{at } r \to 0 .$$
(For other values of $\kappa$, the story is not as simple,
but there is a solution with the same qualitative properties.  See for example
\S3.3 of E.~Weinberg, {\it Classical solutions in Quantum Field Theory}.)

\item The second BPS equation and \eqref{eq:BCatinfty} 
imply that all the action (in particular the support of $F_{12}$) is localized near $r=0$.  
Evaluate the 
magnetic flux through the $x^1-x^2$ plane, 
$\Phi \equiv \int B\cdot d a $ in the vortex configuration labelled by $n$.
Show that the energy density is $ U = {v^2 \over 2} \cdot \Phi $.



%\answer{The tricky bit in the energy is 
%that the second term is 
%\begin{align}
%- \int d^2 x { \half} \ii \eps_{k\ell} \partial_k \( \phi^\star D_\ell\phi\) 
%&= -\half \ii  \int d  \( \phi^\star D_{\ell} \phi dx^\ell \) 
%\buildrel{\text{Stokes}}\over{=}  - \half \ii \oint_{C_\infty} \( \phi^\star D_{\ell} \phi dx^\ell \) 
%\\&= 
%- \half \oint_{C_\infty} A |\phi|^2 = - \half \Phi v^2.
%\end{align}
%}
%

\item Show that the previous result for the flux follows from demanding
that the two terms in $ D_i \phi$ cancel at large $r$ so that 
\be\label{eq:dphi} D_i \phi \buildrel{ r \to \infty}\over{\to} 0 \ee
faster than $1/r$. 
Solve \eqref{eq:dphi} for $A_i$ in terms of $\phi$ and integrate $ \int d^2 x F_{12}$.  


\item Argue that a single vortex (string) in the ungauged  theory 
(with global $\gU(1)$ symmetry) 
$$ \CL = |\partial \phi |^2 + V(|\phi|) $$ 
does not have finite energy per unit length.
By a vortex, I mean a configuration where $ \phi \buildrel{ r \to \infty}\over{\to}  v e^{ \ii \varphi} $,
where $x^1 + \ii x^2 = r e^{ \ii \varphi} $.


\item Consider now the case where the scalar field has charge $q$.
(Recall that in a superconductor made by BCS pairing of electrons,
the charged field which condenses has electric charge two.)
Show that the magnetic flux in the core of the minimal ($n=1$) vortex
is now (restoring units) $ {h c \over q e }$.  
%(This value of the flux was first measured by Fairbanks.)
This is a real thing that people can measure.  





\end{enumerate}


\item {\bf BPS conditions from supersymmetry.} [bonus!]
What's special about $ \kappa = \kappa_0$?  
For one thing, it is the boundary between type I and type II superconductors (which are distinguished by the size of the vortex core).
More sharply, it means the mass of the scalar equals the mass of the vector, 
at least classically.
Moreover, in the presence of some extra fermionic fields, 
the model with this coupling has an additional symmetry
mixing bosons and fermions, namely supersymmetry.
This symmetry underlies the special features we found above.
Here is an outline (you can do some parts without doing others) of how this works.
The logic in part (c) underlies a lot of the progress in string theory
since the mid-1990s.
Please do not trust my numerical factors.

\begin{enumerate}
\item Add to $\CL_h$ a charged fermion $\Psi$ (partner of $\phi$)
and a neutral Majorana fermion $\lambda$ (partner of $A_\mu$): 
$$ \CL_f = { \half } \ii \bar \Psi \slashed{D} \Psi
%+ g \phi \bar \Psi \Psi 
+ \ii \bar \lambda \slashed{D} \lambda +  \bar \lambda \Psi \phi + h.c.   .$$
Consider the transformation rules
$$ \delta_\eps A_\mu = \ii \bar \eps \gamma_\mu \lambda, 
\delta_\eps \Psi =   D_\mu \phi  \gamma^\mu \eps, ~~
\delta_\eps \phi = -  \ii \bar \eps \Psi, 
\delta_\eps \lambda   = - \half \ii \sigma^{\mu\nu} F_{\mu\nu} \eps + \ii ( |\phi|^2 -v ) \eps
$$
where the transformation parameter $ \eps$ is a Majorana spinor
(and a grassmann variable).
Show that (something like this) is a symmetry of $\CL = \CL_h + \CL_f$.
This is $\CN=1$ supersymmetry in $D=4$.

\item Show that the conserved charges 
associated with these transformations $Q_\alpha$ 
(they are grassmann objects and spinors, since 
they generate the transformations, via $ \delta_\eps \text{fields} = [ \eps_\alpha   Q_\alpha + h.c., \text{fields}]$),
satisfy the algebra
\be\label{eq:susy!}  \{ Q, \bar Q \} = 2 \gamma^\mu P_\mu  + 2 \gamma^\mu \Sigma_\mu \ee
where $P_\mu$ is the usual generator of spacetime translations and
$ \Sigma_\mu$ is the {\it vortex string charge},
which is nonzero in a state with a vortex string stretching in the $\mu$ direction.
$\bar Q \equiv Q^\dagger \gamma^0$ as usual.
%$$ \Sigma_\mu 

\item If we multiply \eqref{eq:susy!} on the right by $ \gamma^0$, 
we get the positive operator
$ \{ Q_\alpha, Q^\dagger_\beta\} $.
This operator annihilates states which satisfy 
$ Q \ket{BPS}=0$ for some components of $Q$. 
Such a state is therefore invariant under some subgroup of the superymmetry,
and is called a BPS state.  
Now look at the right hand side of \eqref{eq:susy!}$ \times \gamma^0$
in a configuration where $ \Sigma_3 = \pi n v^2$
and show that its energy density is $ E \ge \pi |n|v^2$, 
with the inequality saturated only for BPS states.

\item To find BPS configurations, we can simply set to zero
the relevant supersymmetry variations of the fields.  
Since we are going to get rid of the fermion fields anyway, we can set them to zero
and consider just the (bosonic) variations of the fermionic fields.
Show that this reproduces the BPS equations.

This line of thought is the crucial ingredient by which 
progress has been made in understanding many supersymmetric theories, including superstring theories.  
For more, see for example \htmladdnormallink{here}{https://arxiv.org/abs/hep-th/9407087}
or chapter 14 of Polchinski's book.


\end{enumerate}




\end{enumerate}
\end{document}